Affinity-based clustering of vectors for partitioning the columns of a matrix

ABSTRACT

A method and computer program product for partitioning the columns of a matrix A. The method includes providing the matrix A in a memory device of a computer system. The matrix A has n columns and m rows, wherein n is an integer of at least 3, and wherein m is an integer of at least 1. The method further includes executing an algorithm by a processor of the computer system. Executing the algorithm includes partitioning the n columns of the matrix A into a closed group of p clusters, wherein p is a positive integer of at least 2 and less than n, wherein the partitioning includes an affinity-based merging of clusters of the matrix A, and wherein each cluster is a collection of one or more columns of A.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates to a method for partitioning the columnsof a matrix.

2. Related Art

Critical dimensions in microelectronic designs have been rapidly scalingto smaller and smaller values leading to increased parametricvariability in digital integrated circuit performance. The resultingincrease in the number of significant and independent sources ofvariation leads to exponential complexity for traditional static timingmethodologies. One solution to this problem is path-based statisticaltiming analysis, in which the probability distribution of theperformance of a chip is computed in a single analysis, simultaneouslytaking into account all sources of variation. Such probabilistic methodsare often dependent on restricting the sources of variation to a smallnumber, and several algorithms proposed in the literature haveexponential complexity in the dimensionality of the process space. Thus,there is a need for a method that facilitates an efficient and simpleuse of statistical timing analysis in a way that takes into account theeffect of all pertinent sources of variation.

SUMMARY OF THE INVENTION

The present invention provides a method of partitioning the columns of amatrix A, said method comprising:

providing the matrix A in a memory device of a computer system, saidmatrix A having n columns and m rows, n being an integer of at least 3,m being an integer of at least 1; and

executing an algorithm by a processor of the computer system, saidexecuting including partitioning the n columns of the matrix A into aclosed group of p clusters, p being a positive integer of at least 2 andless than n, said partitioning comprising an affinity-based merging ofclusters of the matrix A, each said cluster consisting of one or morecolumns of said matrix A.

The present invention provides a method of partitioning the columns of amatrix A, said method comprising executing an algorithm by a processorof a computer system, said executing including performing the steps of:

generating a list of clusters having n clusters such that each of the nclusters consist of a unique column of the matrix A, said matrix A beingstored in a memory device of the computer system, said matrix A having ncolumns and m rows, n being an integer of at least 3, m being an integerof at least 1, each said cluster consisting of one or more columns ofsaid matrix A;

determining if a termination condition is satisfied and if saiddetermining so determines that said termination condition is satisfiedthen terminating said executing else performing the following steps:

-   -   selecting a next pair of clusters from the list of clusters,        said next pair of clusters consisting of a first cluster and a        second cluster, said next pair of clusters having an affinity        that is not less than an affinity between any pair of clusters        not yet selected from the list of clusters;    -   merging the first and second clusters to form a new cluster;    -   inserting the new cluster into the list of clusters while        removing the first and second clusters from the list of        clusters; and    -   re-executing said determining step.

The present invention provides a computer program product, comprising acomputer usable medium having a computer readable program embodiedtherein, said computer readable program comprising an algorithm forpartitioning the columns of a matrix A, said algorithm adapted toperform the steps of:

providing the matrix A in a memory device of the computer system, saidmatrix A having n columns and m rows, n being an integer of at least 3,m being an integer of at least 1; and

partitioning the n columns of the matrix A into a closed group of pclusters, p being a positive integer of at least 2 and less than n, saidpartitioning comprising an affinity-based merging of clusters of thematrix A, each said cluster consisting of one or more columns of saidmatrix A.

The present invention provides a computer program product, comprising acomputer usable medium having a computer readable program embodiedtherein, said computer readable program comprising an algorithm forpartitioning the columns of a matrix A, said algorithm adapted toperform the steps of:

generating a list of clusters having n clusters such that each of the nclusters is a unique column of the matrix A, said matrix A being storedin a memory device of the computer system, said matrix A having ncolumns and m rows, n being an integer of at least 3, m being an integerof at least 1, each said cluster consisting of one or more columns ofsaid matrix A;

determining if a termination condition is satisfied and if saiddetermining so determines that said termination condition is satisfiedthen terminating said algorithm else executing the following steps:

-   -   selecting a next pair of clusters from the list of clusters,        said next pair of clusters consisting of a first cluster and a        second cluster, said next pair of clusters having an affinity        that is not less than an affinity between any pair of clusters        not yet selected from the list of clusters;    -   merging the first and second clusters to form a new cluster;    -   inserting the new cluster into the list of clusters while        removing the first and second clusters from the list of        clusters; and    -   re-executing said determining step.

The present invention advantageously facilitates an efficient and simpleuse of statistical timing analysis of electrical circuits in a way thattakes into account the effect of all pertinent sources of variation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of a path-based statistical timing methodincluding a dimensionality reduction, in accordance with embodiments ofthe present invention.

FIG. 2 is a flow chart describing the dimensionality reduction of FIG. 1including a matrix partitioning, in accordance with embodiments of thepresent invention.

FIG. 3 is a plot of affinity versus alignment for different values ofnorm-ratio, in accordance with embodiments of the present invention.

FIG. 4 is a flow chart describing the matrix partitioning of FIG. 2, inaccordance with embodiments of the present invention.

FIG. 5 illustrates a computer system used for affinity-based clusteringof vectors for partitioning the columns of a matrix, in accordance withembodiments of the present invention

DETAILED DESCRIPTION OF THE INVENTION

The present invention discloses a method for partitioning the columns ofa matrix on the basis of a novel affinity measure. One application ofthe inventive method is reducing the number of independent variablesutilized in a linear model, through a dimensionality-reduction schemethat uses a singular value decomposition (SVD) in conjunction withpartitioning of the independent variables. Another application of theinventive method is that of a statistical timer which could thenefficiently analyze the reduced-dimension system.

FIG. 1 is a flow chart of a path-based statistical timing method inapplication to semiconductor chip timing, in accordance with embodimentsof the present invention. A path is defined as a sequence of circuitelements through which a signal propagates. A complete path may include:the launching clock path, the data path, and the capturing clock path,so that correlations and commonalities between data and correspondingclock paths are fully captured.

A path slack is defined as the difference in time between a requiredarrival time (the latest or earliest time a signal must be stable inorder or for the circuit to operate properly) of a signal at an endpoint P in the path and a calculated or actual arrival time (the latestor earliest time a signal actually becomes stable) of the signal at thepoint P. For sequential circuits, data arriving at a latch typicallymust be stable before the clock transition, and thus the path slack maybe viewed as the algebraic difference between a clock path delay and adata path delay. The timing analysis method of FIG. 1 verifies theacceptability of the timing of the designed chip by confirming, forexample, that the magnitude of the path delay is less than the clockcycle time so that the chip could be reliably operated. The timinganalysis method calculates path slacks for each of m paths, wherein m isa positive integer. The m paths need not represent all timing paths inthe chip, but may represent the m nominally most critical paths or acollection of the nominally most critical path or paths at each endpoint. As many paths are collected as are deemed to have a significantchance of being critical at some point in the process space.

The first input is a netlist 12, which describes the structure andtopology of the circuit to be analyzed. The timing analysis method usestiming assertions 14 and timing rules 16. Examples of timing assertions14 include, inter alia, arrival times at the inputs, required arrivaltimes at the outputs, external loads driven by the outputs, and clockedge and phase information. Examples of timing rules 16 include, interalia, delay models for the delay and slew (rise/fall time) of everyinput-to-output transition of every component of the electrical circuit.These delay models may be in the form of equations withpre-characterized coefficients, or table models or means for computingthe delay models on the fly by means of simulation.

The method initially uses a well-known Static Timing Analysis 10 tocalculate nominal path slacks for each of m paths, wherein m is apositive integer. The m nominal path slacks are characterized by avector dε

^(m) (i.e., the m-dimensional space of real numbers) and are calculatedby using nominal values of the independent variables modeled.

After the Static Timing Analysis 10 is performed, the method executes anm Paths Selection Correlation Computation 20, utilizing a processvariation vector x 22 (described infra). The Computation 20 may includea static timing analysis with forward propagation of nominal arrivaltimes and backward propagation of nominal required arrival times througha timing graph representing the timing properties of the electricalcircuit. It may also include a path-tracing procedure to list thenominally most critical paths, or the most critical path or pathsleading up to each end point. It may also include a slack recordingprocedure to record the nominal slack of each such path. It may alsoinclude a sensitivity procedure to determine the sensitivity of eachsuch path slack to each source of variation. The sensitivity proceduremay traverse the path and collect sensitivities from the delay models,or may use simulation on the fly, or may employ finite differencing todetermine the sensitivities.

The Static Timing Analysis 10 produces a nominal result of the timing ofthe circuit, whereas the present invention determines the fullprobability distribution of the timing of the circuit while taking intoaccount sources of variation. With the Static Timing Analysis 10, thenominal calculated arrival time of the signal at the point P does notaccount for the dependence of path slack on a set of independentstochastic variables whose deviation from nominal values is described bya vector xε

^(n) (i.e., the vector x 22 of FIG. 1) of order n having elements x_(i)(i=1, 2, . . . , n). Each independent variable x_(i) is a randomvariable characterized by a probability density function (PDF), whichmay represent a normal probability distribution or any other form ofprobability distribution.

The independent variables x_(i) may be, inter alia, process variationswhich may fit into some or all of the following categories: processvariations (e.g., control parameters operative during processing such asfocus of a lens during photolithography or dose of ions duringimplantation); manufacturing variations (e.g., manufacturing dimensionaltolerances), operating conditions variations (e.g., operatingtemperature of the chip, power supply voltage); fatigue-relatedvariations (e.g., power-on hours or product lifetime); and modelingerrors (e.g., neglect or approximation of a term in a modelingequation); material property variations (e.g., electricalconductivities, capacitances, doping concentrations, etc.).

The path slack variations d of the paths are each assumed to be a linearfunction of the sources of variation x_(i). Although, the assumption oflinearity may represent an approximation for one or more of thevariables x_(i), the linearity approximation for any variable x_(i) canbe made increasingly accurate by decreasing the numerical range ofx_(i). The path slacks are modeled asd+d= d+Ax  (1)wherein dε

^(m) represents the variation of the path slacks from their nominalvalues and Aε

^(m×n) is a coefficient matrix which facilitates expression of the pathslack variations d as a linear combination of the process variations x.The matrix A is identified in FIG. 1 as the coefficient matrix A 24.Scaling may be applied in a pre-processing step to obtain circularsymmetry; i.e., the x variables x_(i) can be treated, for convenienceand without loss of generality, as unit-variance zero-mean independentrandom variables after the scaling.

The dimensionality reduction step 30 generates a new coefficient matrixB and an associated new set of p statistically independent variables ina vector z of order p, wherein p<n. The new coefficient matrix B isidentified in FIG. 1 as the new coefficient matrix B 34, wherein B is anm×p matrix. The new process variable vector z, which is associated withthe new coefficient matrix B, is identified in FIG. 1 as the vector z ofreduced variables 32. Implementation of the dimensionality reductionstep 30 will be discussed infra in conjunction with FIGS. 2-4.

Following the dimensionality reduction step 30, a statistical timinganalysis 36 may be performed as described, for example, by Jess et al.(see Reference [1]). A yield curve 38 may be determined as an output ofthe timing analysis 36. Various diagnostics may also be produced as anoutput of the timing analysis 36. The efficiency of all three methodswould benefit from performing the analysis in a lower-dimensional space.

FIG. 2 is a flow chart describing the dimensionality reduction 30 ofFIG. 1, in accordance with embodiments of the present invention. Theprocess of FIG. 2 starts with the m×n matrix A 24 and vector x 22 oforder n, and determines the m×p matrix B 34 and the vector z 32 of orderp, wherein 2≦p<n. As p/n is reduced, further dimensionality reduction isachieved. However, decreasing p relative to n increases the numericalerrors of the procedure, so that a lower bound for p is controlled by adegree of accuracy desired, as will be explained infra.

The process of FIG. 2 starts with the linear model of the deviation ofthe path slacks from their nominal valuesd=Ax  (2)The number of paths m may be large (such as, inter alia, 10,000 orhigher), while the number of statistically independent process variablesn may be smaller (such as, inter alia, 30 to 50). The method of FIG. 2seeks a reduction in the number of statistically independent processvariables by looking for a new set of statistically independent processvariables zε

^(p), where 2≦p<n (n≧3), such that{tilde over (d)}=Bz  (3)where B is the new m×p coefficient matrix, and {tilde over (d)}ε

^(m) is the new path-slack variation vector, which is thelower-dimensionality approximation to the original vector d. The number(p) of new process variables may be small (such as, inter alia, 2 to 5).Moreover, each new process variable will be a linear combination of theold process variables. For some choices of B, however, z may not bestatistically independent, especially in the case of non-Gaussianprocess variations. However, Li et al. (see Reference [2]) have used atechnique based on Singular Value Decomposition (SVD) to find B and zsuch that: the new process variables z are also statisticallyindependent, even for non-Gaussian sources of variation; and theEuclidean norm of the error ∥d−{tilde over (d)}∥ is as small aspossible. Li et al. have shown that statistical independence ispreserved by partitioning the columns of the original coefficient matrixA into p clusters and constructing the new coefficient matrix B and thenew process variables z using the (largest) singular values andcorresponding (left and right) singular vectors of each of the pclusters of A in a technique referred to as SMSVD (Sub-Matrix SVD).Accordingly, step 40 of FIG. 2 partitions the columns of the coefficientmatrix A in top clusters denoted as A₍₁₎, A₍₂₎, . . . , and A_((p)). Thenovel manner in which the coefficient matrix A is partitioned into the pclusters in accordance with the present invention is disclosed herein inaccordance with FIGS. 3-4 and a description thereof.

Assuming that the coefficient matrix A has been partitioned into the pclusters (as will be described infra), let A_((k)) denote the clustercorresponding to the k-th cluster and let x_((k)) denote thecorresponding sub-vector of process variables. Using SVD (see Reference[3]), step 42 of FIG. 2 approximates each cluster A_((k)) by itsprincipal component asA _((k))≈σ_((k),1) u _((k),1) v _((k),1) ^(T)  (4)where σ_((k),1) is the largest singular value, u_((k),1) is thecorresponding left singular vector, and v_((k),1) is the correspondingright singular vector of the k-th cluster A_((k)). Equation (4) is theonly source of approximation in the dimensionality reduction procedurerelating to the SMSVD technique of Reference [2].

The k-th column of the new m×p coefficient matrix B isb _(k)=σ_((k),1) u _((k),1)  (5)which is the left singular vector of the cluster A_((k)) of the k-thcluster scaled by its largest singular value. The k-th new processvariable z_(k) is:z _(k) =v _((k),1) ^(T) x _((k))  (6)Accordingly, step 44 of FIG. 2 determines b_(k) and z_(k) via Equations(5) and (6), respectively. Since z_(k) depends only on original processvariables in the k-th cluster and the different clusters are disjoint bydefinition, statistical independence is preserved in z, as shown inReference [2]. Since each of these (left or right) singular vectors hasa unit norm, the linear transformation (6) ensures that the z variablesalso have zero mean and unit variance, given that the original xvariables had zero mean and unit variance, which is an addedconvenience.

The affinity-based clustering technique of the present invention willnext be described. As remarked earlier, the only approximation in thedimensionality reduction procedure is in Equation (4). From SVD theory(See Reference [3]), it can be shown that the cluster approximationerror E in the SVD approximation can be measured by

$\begin{matrix}{E = {\max\limits_{{k = 1},2,{\ldots\; p}}\left( \frac{\sigma_{{(k)},2}}{\sigma_{{(k)},1}} \right)}} & (7)\end{matrix}$which means that for each cluster, as long as the second largestsingular value σ_((k),2) of its cluster A_((k)) is much smaller than itslargest singular value σ_((k),1), there will be little or no error dueto the approximation of Equation (4). For any given cluster of columnsof the matrix A, one can compute the approximation error by Equation (7)above. A brute-force method of trying all possible clusters and pickingthe best having the smaller error can be used, but is impractical forlarge n, since Reference [4] shows that the total number of ways topartition n objects into p clusters is given by the Stirling Number ofthe second kind which is exponential in n.

However, the present invention discloses an algorithm to find a goodclustering of columns of A, that will result in a small error computedby Equation. (7) as will be discussed next. Specifically, the novelclustering technique of the present invention exploits a novel affinitymeasure between two vectors, wherein the affinity is a function of theratio of the norms of the two vectors and of a relative alignmentbetween the two vectors. This clustering scheme is based on thepair-wise affinity between the columns of A. To this end, let w and y beany two arbitrary non-zero vectors. Assuming that ∥y∥≦∥w∥, thenorm-ratio α between y and w is defined as

$\begin{matrix}{\alpha = \frac{y}{w}} & (8)\end{matrix}$and the alignment β between y and w is defined as

$\begin{matrix}{\beta = \frac{{y^{T}w}}{{y}\mspace{11mu}{w}}} & (9)\end{matrix}$so that the alignment β may be interpreted as the cosine of the anglebetween vectors y and w. The norm ∥y∥ is defined as the root mean squareof the entries of y; i.e., if y is a vector of order n having orderedcomponents y₁, y₂, y₃, . . . y_(n), then∥y∥=[(y ₁)²+(y ₂)²+(y ₃)²+ . . . +(y _(n))²]^(1/2)

It is assumed herein that 0<α≦1 and it follows from the Cauchy-Schwartzinequality that 0≦β≦1. Also, if β=0 then the two vectors are orthogonal(i.e., the angle between y and w is either 90 or 270 degrees). If β=1,then y and w are perfectly aligned (i.e., the angle between y and w iseither 0 or 180 degrees).

Next, effect of clustering y and w together is considered by defining amatrix S=[wy] with 2 columns w and y, and computing the two (largest)singular values of S. Note that the further the gap is between these twosingular values, the better is the quality of clustering y in the samepartition as w. Note that a “partition,” a “cluster,” and a “sub-matrix”each have the same meaning herein. Thus a cluster of the matrix A isdefined herein, including in the claims, as a matrix that includes oneor more columns of the matrix A such that no column of the matrix Aappears more than once in the cluster. A group of two or more clustersof the matrix A is defined herein and in the claims to be a “closedgroup” of the matrix A if no two clusters in the group include a samecolumn of the matrix A and if the group of clusters collectively includeall columns of the matrix A. By the preceding definition, any column orgroup of columns of the matrix A is a cluster of the matrix A. Anycluster not in the closed group is said to be outside of the closedgroup. The present invention, however, discloses infra how to formparticular clusters of the matrix A based on using an “affinity” of thecolumns of the matrix A, wherein said particular clusters of the matrixA are utilized in a manner that reduces or minimizes the error expressedin Equation (7).

Forming S^(T)S:

${S^{T}S} = {\begin{bmatrix}{w^{T}w} & {w^{T}y} \\{y^{T}w} & {y^{T}y}\end{bmatrix} = {{w}^{2}\begin{bmatrix}1 & {\pm {\alpha\beta}} \\{\pm {\alpha\beta}} & \alpha^{2}\end{bmatrix}}}$Thus, the eigen-values of S^(T)S are ∥w∥² times the roots λ₁ and λ₂ ofthe characteristic polynomial derived as follows:

${\det\begin{bmatrix}{\lambda - 1} & {\mp {\alpha\beta}} \\{\mp {\alpha\beta}} & {\lambda - \alpha^{2}}\end{bmatrix}} = {{{\left( {\lambda - 1} \right)\left( {\lambda - \alpha^{2}} \right)} - {\alpha^{2}\beta^{2}}}\mspace{205mu} = {\lambda^{2} - {\left( {1 + \alpha^{2}} \right)\lambda} + {\alpha^{2}\left( {1 - \beta^{2}} \right)}}}$so  that  $\lambda_{1,2} = \frac{\left( {1 + \alpha^{2}} \right) \pm \sqrt{\left( {1 + \alpha^{2}} \right)^{2} - {4{\alpha^{2}\left( {1 - \beta^{2}} \right)}}}}{2}$But the singular values σ₁(S) and σ₂(S) of S are ∥w∥ times thesquare-root of λ₁ and λ₂, respectively, which are:σ₁(S)=∥w∥√{square root over (λ₁)}, σ₂(S)=∥w∥√{square root over (λ₂)}The affinity between the two vectors y and w is now defined to be

$\begin{matrix}{{{affinity} = {{1 - \frac{\sigma_{2}(S)}{\sigma_{1}(S)}}\mspace{76mu} = {{1 - \frac{\sqrt{\lambda_{2}}}{\sqrt{\lambda_{1}}}}\mspace{79mu} = {1 - \sqrt{\frac{1 + \alpha^{2} - \sqrt{\left( {1 + \alpha^{2}} \right)^{2} - {4{\alpha^{2}\left( {1 - \beta^{2}} \right)}}}}{1 + \alpha^{2} + \sqrt{\left( {1 + \alpha^{2}} \right)^{2} - {4{\alpha^{2}\left( {1 - \beta^{2}} \right)}}}}}}}}}\;} & (10)\end{matrix}$

Using Equation (10), FIG. 3 shows the plot of affinity versus thealignment β for several values of the norm-ratio α. From Equation (10),the affinity is a real number between 0 and 1. For each of the curves ofFIG. 3, the affinity at β=1 (i.e., perfect alignment between y and w) is1 while the affinity at β=0 (i.e., y and w are orthogonal to each other)is 1−α. The affinity is a monotonically increasing function of thealignment β for a fixed norm-ratio α. The closer the affinity is to 1,the better is the quality of the cluster obtained by putting y in thesame cluster as w. An affinity close to 0 means y and w should be indifferent clusters. This explains the clustering algorithm depicted inFIG. 4 for clustering the n columns of a given m×n matrix A into pparts, assuming 2≦p<n (n≧3).

In FIG. 4, the input to the algorithm comprises the coefficient matrix A(denoted as reference numeral 24) and specification of the reducednumber p of independent process variables (denoted as reference numeral26). Note that the matrix A is provided in a memory device of a computersystem such as the computer system 90 of FIG. 5 discussed infra. A meansfor providing the matrix A in the memory device may include, inter alia,reading the matrix A into the memory device.

The algorithm depicted in FIG. 4 comprises steps 51-60.

Step 51 computes the norm-ratio α, alignment β, and the affinity, usingEquations (8), (9), and (10) for each pair of columns in the coefficientmatrix A.

Step 52 sorts the column pairs in decreasing order of affinity.

Step 53 creates a list of clusters comprising n initial clusters suchthat each of the n initial clusters is a unique column of thecoefficient matrix A. Generally, a cluster includes one or more columnsof the coefficient matrix A as will be seen infra through the clustermerging step 59. Each cluster has a “leader” which is the column havingthe largest norm of all of the columns in the cluster. Thus initially,every column of the coefficient matrix A is both a cluster and a leaderof the cluster, and the initial number of clusters is equal to n. Notethat the affinity between two clusters is defined as the affinitybetween the leaders of the two clusters.

Step 54 initializes a cluster count index q to n (i.e., q=n initially).

Step 55 determines whether to terminate the clustering algorithm basedon satisfaction of a termination condition. In one embodiment, thetermination condition is that the number of clusters formed thus far (q)is equal to the desired number of reduced independent process variables(p). Alternative termination conditions will be discussed infra. If YES(i.e., q=p) then the process STOPs in step 56. If NO (i.e., q>p) thenstep 57 is next executed.

Step 57 selects a next pair of clusters having an affinity therebetween(i.e., between the two clusters of the next pair of clusters) not lessthan an affinity between clusters of any existing pair of clusters notyet selected from the list of clusters. During the first execution ofstep 57, the existing clusters in the list of clusters each consist of asingle column as explained supra in conjunction with step 53, so thatthe next pair of clusters is selected from the sorted list of columnpairs. It should be recalled from step 52 that the column pairs arepositioned in the sorted list of column pairs in order of decreasingvalue of affinity, so the first column pair picked has the highestaffinity; the second column pair to be picked has the second highestaffinity, etc.

The pair of clusters selected in step 57 consists of two clusters,namely a first cluster and a second cluster. Step 58 determines, for thecluster pair picked in step 57, whether to merge the first cluster withthe second cluster to form a new cluster. During the first execution ofstep 58, the first and second clusters of the cluster pair selected instep 57 respectively consist of a first column and a second column asexplained supra in conjunction with step 53, and step 58 determineswhether the first column and the second column from the column pair areeach in a separate cluster (i.e., whether the first and second clustersare different clusters), and if so whether the first and second columnsare leaders of the first and second clusters, respectively. If the firstcolumn and second columns are in the same cluster already or if eitherthe first or second column is not a leader currently of its cluster,then the procedure returns to step 57 to select the next pair ofclusters. Otherwise, step 59 is next executed.

Step 59 merges the first and second clusters to form a new cluster. LetN₁ and N₂ represent the norm of the leader of the first and secondclusters, respectively. The leader of the new cluster is the leader (L₁)of the first cluster if N₁>N₂ or the leader of the new cluster is theleader (L₂) of the second cluster if N₂>N₁. If N₁=N₂ then either L₁ orL₂ could be the leader of the new cluster, using an arbitrary choice ofL₁ or L₂, or a random selection of L₁ or L₂. The new cluster thus formedis added to the list of clusters and the first and second clusters areremoved from the list of clusters.

In an embodiment, the merging of the first and second clusters to formthe new cluster in step 59 removes the first and second clusters asindividual clusters for subsequent consideration in the partitioning ofthe columns of the matrix A. This also removes any non-leader columnpair in the list of column pairs of step 52, wherein said any non-leadercolumn pair includes any column of the merged first and second clustersthat is not the leader of the merged cluster. Thus in this embodiment,step 58 is unnecessary since the conditions for not merging the clustersof the selected cluster pair appear cannot be satisfied; i.e., the firstand second columns will always be a leader in the case of a clusterconsisting of more than one column, and the first and second clusters ofthe cluster pair must be in different clusters due to said removal ofcolumn pairs in the list of cluster pairs. Hence in this embodiment,each pair of clusters selected in step 57 will be merged to form a newcluster in step 59.

The procedure next executes step 60, which decrements the cluster countindex q by 1, inasmuch as two clusters have been merged into a singlecluster (i.e., the new cluster in step 59). The procedure next returnsto step 55.

Note that the particular clusters formed by the process of FIG. 4 mayconsist of merged clusters and unmerged clusters. The merged clustersare formed in step 59. The unmerged clusters (if any) consist of anyclusters of the initially formed n clusters in step 53 that have notbeen merged in step 59. As an example, consider an example of n=5 andp=3, wherein the group of clusters formed by the affinity-based processof FIG. 4 results in the formation of three clusters consisting of afirst cluster of columns 2 and 5 (a merged cluster), a second cluster ofcolumns 3 and 4 (a merged cluster), and a third cluster of column 1 (anunmerged cluster). Of course, many other clusters of the matrix A alsoexist but are not selected in the affinity-based procedure (e.g., acluster consisting of columns 1 and 3, a cluster consisting of columns2-5, a cluster consisting of column 2, etc.).

Consider the following numerical example to illustrate the clusteringtechnique of the present invention, using the following 6×4 matrix Asuch that m=6 and n=4:

$\begin{matrix}{A = \begin{bmatrix}9 & {7} & 8 & {6} \\9 & {- 7} & 8 & {- 6} \\9 & {7} & 8 & {6} \\9 & {- 7} & 8 & {- 6} \\9 & {7} & 8 & {6} \\9 & {- 7} & 8 & {- 6}\end{bmatrix}} & (11)\end{matrix}$

Note that A is a 6×4 matrix with n=4 columns and m=6 rows. It is desiredto reduce the number of process variables to 2 so that the procedurewill attempt to reduce the initial n=4 clusters to p=2 clusters. First,a list is constructed of all 6 possible pairs of columns sorted indecreasing order of their pair-wise affinity (see step 52 of FIG. 4) asshown in Table 1.

TABLE 1 Column Pair β α Affinity (1, 3) 1 8/9 1 (2, 4) 1 6/7 1 (1, 4) 06/9 1/3 (3, 4) 0 6/8 1/4 (1, 2) 0 7/9 2/9 (2, 3) 0 7/8 1/8Table 1 shows that column pairs (1,3) and (2,4) are each perfectlyaligned (i.e., β=1). This results in an affinity of 1 for these twopairs irrespective of their norm-ratios (α's). Hence, these two pairscome out at the top of the sorted list of column pairs. The remainingpairs are orthogonal (i.e., β=0) resulting in their affinity being 1−α,and hence appear at the bottom of the sorted list. Initially, there are4 clusters (i.e., q=4 initially in accordance with step 54 of FIG. 4),and each column is in its own cluster and is the leader of its cluster(see step 53 of FIG. 4).

Since initially q≠p (i.e., 4≠2), the first pair (1,3) in the sorted listis selected (see steps 55 and 57 of FIG. 4). Both columns 1 and 3 arecurrently in different clusters and leaders of their respectiveclusters, which leads to the “YES” result of decision step 58 of FIG. 4.Therefore step 59 of FIG. 4 merges columns 1 and 3 into a single newcluster {1,3} and marks column 1 as the leader of the new cluster, sincecolumn 1 has a larger norm than the norm of column 3.

Next, step 60 of FIG. 4 decrements the cluster counter q to q=3. Thenext pair in Table 1, which is (2,4), is selected (see step 57 of FIG.4) and both columns 2 and 4 are currently in different clusters andleaders of their respective clusters. Therefore columns 2 and 4 aremerged into a single new cluster {2,4} and column 2 is marked as itsleader (since column 2 has the larger norm), in accordance with steps57-58 of FIG. 4. The cluster counter q is decremented to q=2 (see step60 of FIG. 4) which matches the number p=2, thereby satisfying the STOPcondition (i.e., q=p) of step 55 of FIG. 4, resulting in a final clusterset of {1,3} and {2,4}. This final clustering results in the followingclusters of the original A:

${A_{(1)} = \begin{bmatrix}9 & 8 \\9 & 8 \\9 & 8 \\9 & 8 \\9 & 8 \\9 & 8\end{bmatrix}},\mspace{31mu}{A_{(2)} = \begin{bmatrix}{7} & {6} \\{- 7} & {- 6} \\{7} & {6} \\{- 7} & {- 6} \\{7} & {6} \\{- 7} & {- 6}\end{bmatrix}}$Since each cluster has only two columns, it follows that:

$\begin{matrix}{{A_{(1)}^{T}A_{(1)}} = {{\begin{bmatrix}9 & 9 & 9 & 9 & 9 & 9 \\8 & 8 & 8 & 8 & 8 & 8\end{bmatrix}\begin{bmatrix}9 & 8 \\9 & 8 \\9 & 8 \\9 & 8 \\9 & 8 \\9 & 8\end{bmatrix}} = \begin{bmatrix}486 & 432 \\432 & 384\end{bmatrix}}} & (12) \\{{{{{{A_{(2)}^{T}A_{(2)}} =}\quad}\begin{bmatrix}7 & {- 7} & 7 & {- 7} & 7 & {- 7} \\6 & {- 6} & 6 & {- 6} & 6 & {- 6}\end{bmatrix}}\begin{bmatrix}{7} & {6} \\{- 7} & {- 6} \\{7} & {6} \\{- 7} & {- 6} \\{7} & {6} \\{- 7} & {- 6}\end{bmatrix}} = \begin{bmatrix}294 & 252 \\252 & 216\end{bmatrix}} & (13)\end{matrix}$The eigen-values of the above 2×2 matrices are 870 and 0 for the matrixof Equation (12) and 510 and 0 for the matrix of Equation (13).Therefore, the singular values of the two clusters are:σ_((1),1)=√{square root over (870)}=29.495762, σ_((1),2)=0σ_((2),1)=√{square root over (510)}=22.583180, σ_((2),2)=0Thus, according to Equation (7), the approximation error E for thisparticular choice of clustering is 0, which means that this is the bestcluster partitioning for p=2 for this example.

The corresponding left singular vector u_((k),1) and the right singularvector v_((k),1) of the k-th cluster A_((k)) for the largest singularvalues are calculated as follows. The left singular vector u_((k),1) isan m-component vector computed as being proportional to the leader ofA_((k)). The right singular vector v_((k),1) is a 2-component vectorwhose first element is proportional to the average value of the leaderof A_((k)). Both u_((k),1) and v_((k),1) are renormalized to a unitnorm. Thus, for this example:

${u_{{(1)},1} = {{\frac{1}{\sqrt{6}}\begin{bmatrix}1 \\1 \\1 \\1 \\1 \\1\end{bmatrix}} = \begin{bmatrix}0.4082 \\0.4082 \\0.4082 \\0.4082 \\0.4082 \\0.4082\end{bmatrix}}},\mspace{14mu}{v_{{(1)},1} = {{\frac{1}{\sqrt{9^{2} + 8^{2}}}\begin{bmatrix}9 \\8\end{bmatrix}} = {{\begin{bmatrix}0.7474 \\0.6644\end{bmatrix}u_{{(2)},1}} = {\frac{1}{\sqrt{6}} = {\begin{bmatrix}{1} \\{- 1} \\{1} \\{- 1} \\{1} \\{- 1}\end{bmatrix} = \begin{bmatrix}{0.4082} \\{- 0.4082} \\{0.4082} \\{- 0.4082} \\{0.4082} \\{- 0.4082}\end{bmatrix}}}}}},\mspace{14mu}{v_{{(2)},1} = {{\frac{1}{\sqrt{7^{2} + 6^{2}}}\begin{bmatrix}7 \\6\end{bmatrix}} = \begin{bmatrix}0.7593 \\0.6508\end{bmatrix}}}$Using Equation (5), the two columns b₁ and b₂ of the new coefficientmatrix B are:

$b_{1} = {{\sigma_{{(1)},1}u_{{(1)},1}} = {{\sqrt{145}\begin{bmatrix}1 \\1 \\1 \\1 \\1 \\1\end{bmatrix}} = \begin{bmatrix}12.0416 \\12.0416 \\12.0416 \\12.0416 \\12.0416 \\12.0416\end{bmatrix}}}$ and$b_{2} = {{\sigma_{{(2)},1}u_{{(2)},1}} = {{\sqrt{85}\begin{bmatrix}{1} \\{- 1} \\1 \\{- 1} \\1 \\{- 1}\end{bmatrix}} = \begin{bmatrix}{9.2195} \\{- 9.2195} \\9.2195 \\{- 9.2195} \\9.2195 \\{- 9.2195}\end{bmatrix}}}$Using the computed values of b₁ and b₂, the new coefficient matrix B is:

$B = \begin{bmatrix}12.0416 & {9.2195} \\12.0416 & {- 9.2195} \\12.0416 & 9.2195 \\12.0416 & {- 9.2195} \\12.0416 & 9.2195 \\12.0416 & {- 9.2195}\end{bmatrix}$Using Equation (6), the new process variables z₁ and z₂ are:z ₁ =v _((1),1) ^(T) x ₍₁₎=0.7474x ₁+0.6644x ₃z ₂ =v _((2),1) ^(T) x ₍₂₎=0.7593x ₂+0.6508x ₄

In an embodiment, the desired number of clusters p is supplied as aninput to the algorithm of FIG. 4. In this embodiment the terminationcondition in step 55 of FIG. 4 may be q=p wherein q is the cluster countindex.

In a first alternative embodiment, the above affinity-based clusteringalgorithm stops when the affinity in the current pair of columns fromthe sorted list falls below an affinity threshold. The affinitythreshold may be supplied as an input to the algorithm. If the affinityof the next pair of clusters selected in the selecting step 57 of FIG. 4is less than the affinity threshold, then a flag may be set indicatingthat the termination condition has been satisfied, followed byre-executing step 55 while not executing steps 57-60. This firstalternative embodiment will prevent clustering of columns with pooraffinity. The number of clusters produced by this first alternativeembodiment may be more than the desired number p specified.Alternatively p need not be specified as input for this firstalternative embodiment, resulting in a smallest computed value of p suchthat the affinity in each cluster of the final set of clusters exceedsthe affinity threshold.

In a second alternative embodiment, an error tolerance E could bespecified and the algorithm could be modified to generate the smallestnumber of clusters p for which the cluster approximation error Eaccording to Equation (7) is less than the error tolerance c. The errortolerance ε may be supplied as an input to the algorithm. If theselecting step 57 of FIG. 4 results in list of clusters having a clusterapproximation error E such that E≧ε, then a flag may be set indicatingthat the termination condition has been satisfied, followed byre-executing step 55 while not executing steps 57-60.

In a third alternative embodiment employing a hybrid procedure, assumethat it is desired to partition a matrix H having N columns. In thishybrid procedure, n columns of the N columns are partitioned into pclusters by the affinity-based clustering of the present invention, andthe j columns of the N columns are partitioned into t clusters byanother method (e.g., standard SVD), wherein n+j=N. Values of p and tcould be independent inputs. Alternatively, p+t could be an input,wherein p and t could be determined via any desired criterion such as,inter alia, a criterion of p/t=n/j, i.e., the partition sizes areproportional to the number of columns of the partitions in the originalmatrix, to the nearest integer. As an example of this hybrid procedure,the n columns could be associated with an independent set of processvariables x_(i) (i=1, 2, . . . , n) having non-gaussian sources ofvariation, and the j columns could be associated with j additionalprocess variable wk (k=1, 2, . . . , j) having gaussian sources ofvariation. The combined reduced set of p+t process variables resultingfrom this hybrid procedure would be statistically independent. Thishybrid procedure could be expanded such that n+j<N so that the remainingN−n−j columns of the N columns are independently partitioned into one ormore additional partitions of clusters. Each additional partition ofclusters may be generated by any desired method that preserves thestatistical independence of the final reduced set of process variables.

In summary, the affinity-based clustering of the present inventioncomprises partitioning the n columns of the matrix A into a closed groupof p clusters, p being a positive integer of at least 2 and less than n,said partitioning comprising an affinity-based merging of clusters ofthe matrix A. The following discussion clarifies the meaning of“affinity-based merging” of clusters of the matrix A. Saidaffinity-based merging of clusters of the matrix A considers allcandidate clusters of the matrix A which may be merged pairwise, anddetermines or provides the affinity between the clusters of each suchpair of candidate clusters (“pairwise” affinities). Then theaffinity-based merging selects a pair of such candidate clusters to bemerged based on comparing the pairwise affinities so determined orprovided. In an embodiment described supra, the selected pair ofcandidate clusters was selected because it had a higher affinity thanany other pair of candidate clusters.

Although the inventive affinity-based clustering described herein may beadvantageously applied to path-based statistical timing analysisrelating to semiconductor chip timing, the affinity-based clustering ofthe present invention may be generally used to reduce the number ofdegrees of freedom (i.e., dimensionality) used in any linear model. Forvarious modeling, optimization, and analysis purposes, it may bebeneficial to produce models of reduced dimensionality to capture thesystem behavior as a function of a smaller number of sources ofvariation. Thus, the inventive method may be used in many other domainssuch as, inter alia, mechanical or thermal design. As an example, theperformance (such as vibration, gas mileage, stability or brakingeffectiveness, etc.) of a vehicle or airplane or helicopter can bemodeled as a function of a number of sources of variation such asmachining tolerances, material properties, temperature, etc. As anotherexample, a stochastic model of the power dissipation of sub-circuits ofan integrated circuit can be constructed as a function of temperature,device characteristics, input signals, chip activity, etc, and areduced-dimensionality model can be constructed by the inventive methodand applied in simulation, analysis, optimization or macromodeling ofthe power dissipation.

FIG. 5 illustrates a computer system 90 used for affinity-basedclustering of vectors for partitioning the columns of a matrix, inaccordance with embodiments of the present invention. The computersystem 90 comprises a processor 91, an input device 92 coupled to theprocessor 91, an output device 93 coupled to the processor 91, andmemory devices 94 and 95 each coupled to the processor 91. The inputdevice 92 may be, inter alia, a keyboard, a mouse, etc. The outputdevice 93 may be, inter alia, a printer, a plotter, a computer screen, amagnetic tape, a removable hard disk, a floppy disk, etc. The memorydevices 94 and 95 may be, inter alia, a hard disk, a floppy disk, amagnetic tape, an optical storage such as a compact disc (CD) or adigital video disc (DVD), a dynamic random access memory (DRAM), aread-only memory (ROM), etc. The memory device 95 includes a computercode 97. The computer code 97 includes an algorithm for affinity-basedclustering of vectors for partitioning the columns of a matrix. Theprocessor 91 executes the computer code 97. The memory device 94includes input data 96. The input data 96 includes input required by thecomputer code 97. The output device 93 displays output from the computercode 97. Either or both memory devices 94 and 95 (or one or moreadditional memory devices not shown in FIG. 5) may be used as a computerusable medium (or a computer readable medium or a program storagedevice) having a computer readable program code embodied therein and/orhaving other data stored therein, wherein the computer readable programcode comprises the computer code 97. Generally, a computer programproduct (or, alternatively, an article of manufacture) of the computersystem 90 may comprise said computer usable medium (or said programstorage device).

While FIG. 5 shows the computer system 90 as a particular configurationof hardware and software, any configuration of hardware and software, aswould be known to a person of ordinary skill in the art, may be utilizedfor the purposes stated supra in conjunction with the particularcomputer system 90 of FIG. 5. For example, the memory devices 94 and 95may be portions of a single memory device rather than separate memorydevices.

While embodiments of the present invention have been described hereinfor purposes of illustration, many modifications and changes will becomeapparent to those skilled in the art. Accordingly, the appended claimsare intended to encompass all such modifications and changes as fallwithin the true spirit and scope of this invention.

REFERENCES

-   [1] J. A. G. Jess, K. Kalafala, S. R. Naidu, R. H. J. M. Otten,    and C. Visweswariah, “Statistical timing for parametric yield    prediction of digital integrated circuits,” Proceedings of the 40th    Design Automation Conference, Anaheim, Calif. (June 2003), pages    932-937.-   [2] Z. Li, X. Lu, and W. Shi, “Process Variation Dimension Reduction    Based on SVD,” Proceedings of the IEEE International Symposium on    Circuits and Systems, Bangkok, Thailand (May 2003), Volume IV, pages    672-675.-   [3] G. H. Golub and C. F. Van Loan, Matrix Computations, The John    Hopkins University Press, (1987).-   [4] M. Abramowitz and I. A. Stegun, Handbook of Mathematical    Functions, page 824, New York: Dover Publications, Inc. (1972).

1. A computer program product, comprising a computer usable storagemedium having a computer readable program embodied therein, saidcomputer readable program comprising an algorithm for partitioning thecolumns of a matrix A, said algorithm adapted to perform the steps of:providing the matrix A in a memory device of the computer system, saidmatrix A having n columns and m rows, n being an integer of at least 3,m being an integer of at least 1; partitioning the n columns of thematrix A into a closed group of p clusters, p being a positive integerof at least 2 and less than n, said partitioning comprising anaffinity-based merging of clusters of pairs of clusters of the matrix Abased on an affinity between the clusters in each pair of clusters beingmerged, each said cluster consisting of one or more columns of saidmatrix A, and storing the p clusters in a computer-readable storagedevice.
 2. The computer program product of claim 1, wherein the matrix Arelates a vector x having n elements to a vector d having m elements inaccordance with an equation of d =Ax, wherein the n elements of thevector x consist of n statistically independent variables, and whereinthe m elements of the vector d consist of m dependent variables.
 3. Thecomputer program product of claim 2, wherein after said partitioning thealgorithm is adapted to perform the steps of: computing a vector zhaving p statistically independent elements such that each of the pstatistically independent elements is a linear combination of the nstatistically independent variables; and computing an m×p matrix B fromthe p clusters of the matrix A such that Bz defines a new set of mdependent variables replacing Ax.
 4. The computer program product ofclaim 2, wherein said n statistically independent variables representnon-gaussian sources of variation, wherein the algorithm is furtheradapted to perform the step of selecting the n statistically independentvariables from N statistically independent variables such that N>n, saidN variables consisting of said n variables and a remaining N−nstatistically independent variables, said N−n variables representinggaussian sources of variation.
 5. The computer program product of claim2, said m elements of the vector d denoting path slack variations in asemiconductor chip, said n statistically independent variables denotingsources of statistical error that linearly contribute to said path slackvariations, said sources of statistical error comprising statisticalvariations selected from the group consisting of statistical variationsassociated with processing the semiconductor chip, statisticalvariations associated with manufacturing the semiconductor chip,statistical variations associated with operating the semiconductor chip,statistical variations associated with modeling the semiconductor chip,and statistical variations associated with uncertainties in materialproperties of the semiconductor chip.
 6. A computer program product,comprising a computer usable storage medium having a computer readableprogram embodied therein, said computer readable program comprising analgorithm for partitioning the columns of a matrix A, said algorithmadapted to perform the steps of: generating a list of clusters having nclusters such that each of the n clusters is a unique column of thematrix A, said matrix A being stored in a memory device of the computersystem, said matrix A having n columns and m rows, n being an integer ofat least 2, m being an integer of at least 1, each said clusterconsisting of one or more columns of said matrix A; determining if atermination condition is satisfied and if said determining so determinesthat said termination condition is satisfied then terminating saidalgorithm else executing the following steps: selecting a next pair ofclusters from the list of clusters, said next pair of clustersconsisting of a first cluster and a second cluster, said next pair ofclusters having an affinity that is not less than an affinity betweenany pair of clusters not yet selected from the list of clusters; mergingthe first and second clusters to form a new cluster; inserting the newcluster into the list of clusters while removing the first and secondclusters from the list of clusters; and re-executing said determiningstep, wherein the method further comprises storing the list of clusterscomprising all of said inserted new clusters in a computer-readablestorage device.
 7. The computer program product of claim 6, wherein thealgorithm is adapted to accept an affinity threshold as an input to thealgorithm, wherein if the affinity of the next pair of clusters selectedin the selecting step is less than the affinity threshold then thealgorithm is adapted to execute setting a flag indicating that thetermination condition has been satisfied and again performing thedetermining step while not performing the inserting step.
 8. Thecomputer program product of claim 6, wherein the algorithm is adapted toaccept a cluster error tolerance ε as an input to the algorithm, whereinif the selecting step results in the list of clusters having a clusterapproximation error E such that E≧ε then the algorithm is adapted toexecute setting a flag indicating that the termination condition hasbeen satisfied and again performing the determining step while notperforming the inserting step.
 9. The computer program product of claim6, wherein the matrix A relates a vector x having n elements to a vectord having m elements in accordance with an equation of d=Ax, wherein then elements of the vector x consist of n statistically independentvariables, and wherein the m elements of the vector d consist of mdependent variables.
 10. The computer program product of claim 9, saidalgorithm being adapted to further perform the steps of: computing avector z having p statistically independent elements such that each ofthe p statistically independent elements is a linear combination of then statistically independent variables; and computing an m×p matrix Bfrom the p clusters of the matrix A such that Bz defines a new set of mdependent variables replacing Ax.
 11. The computer program product ofclaim 9, said n statistically independent variables representingnon-gaussian sources of variation, said algorithm further adapted toperform the step of selecting the n statistically independent variablesfrom N statistically independent variables such that N>n, said Nvariables consisting of said n variables and a remaining N−nstatistically independent variables, said N−n variables representinggaussian sources of variation.
 12. The computer program product of claim9, said m elements of the vector d denoting path slack variations in asemiconductor chip, said n statistically independent variables denotingsources of statistical error that linearly contribute to said path slackvariations, said sources of statistical error comprising statisticalvariations selected from the group consisting of statistical variationsassociated with processing the semiconductor chip, statisticalvariations associated with manufacturing the semiconductor chip,statistical variations associated with operating the semiconductor chip,statistical variations associated with modeling the semiconductor chip,and statistical variations associated with uncertainties in materialproperties of the semiconductor chip.